While such a formulation increases the dimension of the regression problem by a factor of \(d\)  (which is potentially quite costly, both in terms of computational time and sample efficiency), it is sometimes more natural and offers several additional benefits. The first is based on the fact that the coordinates used in SINDy often represent the spatial modes of a high-dimensional discretized PDE simulation computed using a dimensionality reduction technique like proper orthogonal decomposition [1]}. Each of these spatial modes have an associated energy \(\lambda _i\)  designated by their singular values. Consequently, the quality of the high-dimensional reconstruction does not depend uniformly on the accuracy of each of the individual spatial modes, but in proportion to their energies. In this setting, we can weight each inner term in the sum of the objective function REF  by the energy of the respective spatial mode to recover the terms which maximize the quality of the full reconstruction, rather than the average dimension-wise reconstruction. Another advantage is instead of having to run parameter tuning on different values of \(k\)  for each system variable, we can specify and cross-validate one global value of the desired sparsity, and the model will automatically determine the correct level of sparsity per dimension.
